Comments on “Vibrations of a mass loaded clamped-free Timoshenko beam”

Journal of Sound and Vibration (1989) 129(2), 343-344

COMMENTS ON “VIBRATIONS OF A MASS-LOADED CLAMPED-FREE TIMOSHENKO BEAM” Bruch and Mitchell [l] recently reported a study of the vibrations of a mass-loaded clamped-free Timoshenko beam. There would appear to be two ways in which their work could be relatively easily and beneficially extended: (i) the beam-mass system considered in reference [l] has one clamped end and a lumped mass and lumped moment of inertia at the other free end, which may be not good enough to simulate a robot arm-the beam base condition is not a real “clamped” condition and the center of the tip body is usually at a certain distance from the beam tip; (ii) the shear coefficient k not only depends on the shape of the cross-section but also on the Poisson ratio, as discussed in references [2,3]. The study can be extended to take account of these points, as follows. The free vibration equation of a Timoshenko beam is given in reference [4] as EI a4u/ax4+pAazu/at2+(p21/kC)

E/kG) a4u/ax2at2=0,

a4u/at4-pl(l+

(1)

in which E is the Young’s modulus, I is the moment of inertia, p is the density of the beam per unit length, A is the cross-section, k is the shear coefficient, and G is the shear modulus. For a free vibration problem, one can assume a solution of the form u(x, t) = U(x) exp (iot).

(2)

Substituting equation (2) into equation (1) gives ~““+2A4L2(8r+82)~“+(4A8L48,&-A4)U=0, in which h4= pAo*/ EI, S, = EI/(2kGAL*) condition

(3)

and S2= 1/(2AL’).

If 6, and S2 satisfy the

A2L2~1+A4L4(81-82)2-A4L4(8,+82)>0,

(4)

then the general solution for U(x) is U(x) = A, cos ax + A2 sin (YX+ A3 cash /3x + A4 sinh px,

(5)

where A, to A4 are constants to be determined, and (Yand /3 are defined as a = [A4L2(S, + S,)+ A*Jl+ A4L4(S, - ij2)*]“*,

~=[-A4L2(S,+S2)+A2~1+A4L4(S,-82)2]”2.

(6)

Since the parameters 6, and S2 depend on Z, A and k, the frequency equation actually depends on the shape of the cross-section (I and A) and S the latter being a function of the Poisson ratio v, as discussed in reference [2]. The boundary conditions for U(x) can be taken as EIU”(0) - Z&U’(O) = 0,

E,,““(O)

+ &U(O)

= 0,

EIU”(L)-(J+Md2)02U’(L)-Mdw*U(L)=O, EIU”‘( L) + A4w2U( L) + Mdw2U’( L) = 0,

(7)

where KR is the rotational spring constant, KL is the translational spring constant, M is the mass of the tip body, J is the moment of inertia of the tip body, and d is the distance between the tip body center and the beam tip. 343 0022-460X/89/050343+02

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@ 1989 Academic

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344

LETTERS TO THE EDITOR

Substituting equation (5) into the four conditions frequency equation -Z&3 -CT

1 -z& e3] e41

e32 e42

1 z#

given by equation (7) yields the

z,p -p

e33

e34

e43

e44

= 0,

(8)

in which e3, = E3s, +A4L4Qc, - h4L4Q&,

,

e33=$s2+A4L4Qc2+h4L4Q~&2,

e32=-d3~,+h4L4Q~1+h4L4QG&,, e34=p3~2+A4L4Q~2+A4L4Q~&2,

e 4,=-&,-A4L4Q&1+A4L4~(~+Q~2)~,, e42 ---~2s,-A4L4Q&,-A4L4~(.i+Q~2)~I, e43=~~2-A4L4Qdc2-A4L4~(~+Q~2)~2, e44=~s2-A4L4Q&2-A4L4~(.?+Qd2)~2, s1 = sin aL, G=cyL,

P=PL,

s2 = sinh PL,

c, = cos aL,

Zn = EI/ KnL,

Q = Ml ~4

c2 = cash BL,

Z, = EI/ KLL3,

J= J/pAL3,

d=d/L.

Equation (8) can be solved by the False Position Iteration Method mentioned in reference

Cl]. W. H. LIU

Department of Mechanical Engineering, Tatung Institute of Technology, Taipei, Taiwan, Republic of China (Received 13 August 1987) REFERENCES

and T. P. MITCHELL 1987Journal of Sound and Vibration 114,341-345. Vibrations of a mass-loaded clamped-free Timoshenko beam. 2. G. R. COWPER 1966 Journal of Applied Mechanics 33, 335-340. The shear coefficient in Timoshenko’s beam theory. 3. J. R. HUTCHINSON 1981 Journal of Applied Mechanics 48, 923-928. Transverse vibrations of beams, exact versus approximate solutions. 4. R. L. BISPLINGHOFF and H. ASHLEY 1962 Pn’nciples of Aeroelasticity. New York: Dover. 1. J.C. BRUCH